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Journal of the Physical Society of Japan

Quasi-Particle Density of States and Thouless Conductance of Disorderedd-wave Superconductors

Bodo Huckestein, Alexander Altland

Published Jan 3, 2003• 0 citations

Abstract

Disorder in electronic systems suppresses diffusion and can lead to localization. The origin of this behavior is enhanced backscattering due to quantum interference. In addition, disorder leads to a broadening of the density of states (DoS). However, the localization properties of a disordered system are usually not discernible from a study of the DoS. It is necessary to calculate two-particle properties or consider the influence of boundary conditions on energy eigenvalues as in the study of Thouless numbers to obtain information about localization. Another feature of disordered systems is the independence of their localization properties from details of the disorder in the system. For example, the range of the disorder affects non-universal quantities like the mean-free path but does not influence whether or not the system shows localization at all. Recently, new symmetry classes came into focus that do not conform to the standard expectations about disordered systems outlined above. As we will show in this contribution, details of disorder do matter for these systems and localization properties leave their mark on the DoS. In particular, we look at a system that presents an idealization of a disordered d-wave superconductor. In our study we neglect effects of self-consistency and concentrate on the d-wave symmetry and influence of different kinds of disorder. We consider the lattice quasi-particle Hamiltonian

DOI: 10.1143/jpsjs.72sa.71

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Disorder in electronic systems suppresses diffusion and can lead to localization. The origin of this behavior is enhanced backscattering due to quantum interference. In addition, disorder leads to a broadening of the density of states (DoS). However, the localization properties of a disordered system are usually not discernible from a study of the DoS. It is necessary to calculate two-particle properties or consider the influence of boundary conditions on energy eigenvalues as in the study of Thouless numbers to obtain information about localization. Another feature of disordered systems is the independence of their localization properties from details of the disorder in the system. For example, the range of the disorder affects non-universal quantities like the mean-free path but does not influence whether or not the system shows localization at all. Recently, new symmetry classes came into focus that do not conform to the standard expectations about disordered systems outlined above. As we will show in this contribution, details of disorder do matter for these systems and localization properties leave their mark on the DoS. In particular, we look at a system that presents an idealization of a disordered d-wave superconductor. In our study we neglect effects of self-consistency and concentrate on the d-wave symmetry and influence of different kinds of disorder. We consider the lattice quasi-particle Hamiltonian

We explore the effects of various kinds of random disorder on the quasiparticle density of states of two-dimensional

d

-wave superconductors using an exact real-space method, incorporating realistic details known about the cuprates. Random on-site energy and pointlike unitary impurity models are found to give rise to a vanishing DOS at the Fermi energy for narrow distributions and low concentrations, respectively, and lead to a finite, but suppressed, DOS at unrealistically large levels of disorder. Smooth disorder arising from impurities located away from the copper-oxide planes meanwhile gives rise to a finite DOS at realistic impurity concentrations. For the case of smooth disorder whose average potential is zero, a resonance is found at zero energy for the quasiparticle DOS at large impurity concentrations. We discuss the implications of these results on the computed low-temperature specific heat, the behavior of which we find is strongly affected by the amount of disorder present in the system. We also compute the localization length as a function of disorder strength for various types of disorder and find that intermediate- and high-energy states are quasiextended for low disorder, and that states near the Fermi energy are strongly localized and have a localization length that exhibits an unusual dependence on the amount of disorder. We comment on the origin of disorder in the cuprates and provide constraints on these based on known results from scanning tunneling spectroscopy and specific heat experiments.

We analyse how the range of disorder affects the localization properties of quasiparticles in a two-dimensional d-wave superconductor within the standard non-linear σ-model approach to disordered systems. We show that for purely long range disorder, which only induces intra-node scattering processes, the approach is free from the ambiguities which often beset the disordered Dirac-fermion theories, and gives rise to a Wess–Zumino–Novikov–Witten action leading to vanishing density of states and finite conductivities. We also study the crossover induced by inter-node scattering due to a short range component of the disorder, thus providing a coherent non-linear σ-model description in agreement with all the various findings of different approaches.

We present a numerical study of the quasi-particle density of states (DoS) of two-dimensional d-wave superconductors in the presence of smooth disorder. We find power law scaling of the DoS with an exponent depending on the strength of the disorder and the superconducting order parameter in quantitative agreement with the theory of Nersesyan et al. (Phys.Rev.Lett. 72, 2628 (1994)). For strong disorder a transition to a constant DoS occurs. Our results are in contrast to the case of short-ranged disorder.

The quantum interference effect on the quasiparticle density of states (DOS) is studied with the diagrammatic technique in two-dimensional d-wave superconductors with dilute nonmagnetic impurities both near the Born and near the unitary limits. We derive in details the expressions of the Goldstone modes (cooperon and diffuson) for quasiparticle diffusion. The DOS for generic Fermi surfaces is shown to be subject to a quantum interference correction of logarithmic suppression, but with various renormalization factors for the Born and unitary limits. Upon approaching the combined limit of unitarity and nested Fermi surface, the DOS correction is found to become a

δ

-function of the energy, which can be used to account for the resonant peak found by the numerical studies.

The kernel polynomial method, as an approach to deriving highly-accurate spectral properties of large sparse matrices, is extended to investigate the effects of disorder in two-dimensional inhomogeneous superconductors. Being able to solve the Bogoliubov-de Gennes equations for quite large square lattices self-consistently, we gain new microscopic insights and make intuitive observations of the localization of low-energy quasiparticles on the nanometer scale. We also compare the behaviors of the optical conductivities of superconductors with different pairing symmetries. We find that the Drude weight turns out to be finite only for the weakly-disordered d -wave superconductor while obvious suppression of the optical conductivity in the infrared range can be observed in the disordered s -wave superconductor.

Within numerically exact solutions of the Bogoliubov-de Gennes equations, we demonstrate that discrepancies between predicted low-energy quasiparticle properties in disordered 2D d-wave superconductors occur because of the unanticipated importance of disorder model details and normal state particle-hole symmetry. For the realistic case, which is best described by a binary alloy model without particle-hole symmetry, we predict density-of-state suppression below an energy scale which appears to be correlated with the corresponding single impurity resonance.